Of course! Let's work through the problem step by step.
### Part A: Factoring Out the Greatest Common Factor (GCF)
Given expression:
[tex]\[ 4x^{10} - 64x^2 \][/tex]
1. Identify the GCF of the coefficients and variables:
- The coefficients are 4 and -64. The GCF of 4 and 64 is 4.
- For the variable [tex]\( x \)[/tex], the terms have [tex]\( x^{10} \)[/tex] and [tex]\( x^2 \)[/tex]. The GCF of [tex]\( x^{10} \)[/tex] and [tex]\( x^2 \)[/tex] is [tex]\( x^2 \)[/tex].
2. Factor out the GCF from the entire expression:
- The GCF of the expression is [tex]\( 4x^2 \)[/tex].
3. Rewrite the expression by factoring out [tex]\( 4x^2 \)[/tex]:
[tex]\[
4x^{10} - 64x^2 = 4x^2 (x^8 - 16)
\][/tex]
So, the expression factored by its GCF is:
[tex]\[ 4x^2 (x^8 - 16) \][/tex]
### Part B: Factoring the Entire Expression Completely
Now, we need to factor the expression [tex]\( x^8 - 16 \)[/tex] completely. Notice that this can be factored further using known algebraic identities and factoring techniques.
1. Recognize [tex]\( x^8 - 16 \)[/tex] as a difference of squares:
- We know [tex]\( x^8 - 16 = (x^4)^2 - (4)^2 \)[/tex].
- Use the difference of squares formula: [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
- Applying it to [tex]\( x^8 - 16 \)[/tex]:
[tex]\[
x^8 - 16 = (x^4 - 4)(x^4 + 4)
\][/tex]
2. Factor each resulting term further if possible:
- For [tex]\( x^4 - 4 \)[/tex]:
[tex]\[
x^4 - 4 = (x^2)^2 - (2)^2 = (x^2 - 2)(x^2 + 2)
\][/tex]
- For [tex]\( x^4 + 4 \)[/tex]: This is a sum of squares and can be factored using complex roots:
[tex]\[
x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2)
\][/tex]
3. Combine all factored terms:
- We now have:
[tex]\[
x^8 - 16 = (x^2 - 2)(x^2 + 2)(x^2 + 2x + 2)(x^2 - 2x + 2)
\][/tex]
4. Therefore:
[tex]\[
4x^{10} - 64x^2 = 4x^2 (x^2 - 2)(x^2 + 2)(x^2 + 2x + 2)(x^2 - 2x + 2)
\][/tex]
So, the completely factored expression is:
[tex]\[ 4x^2 (x^2 - 2)(x^2 + 2)(x^2 + 2x + 2)(x^2 - 2x + 2) \][/tex]
This completes the factoring process.
